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Full text of "Nuclear Physics by Enrico Fermi: Chapter 3"

Chapter II± ALP] 'A PARTICLE EMISSION A Q,^h„, A- Penetration of Pen tangular Earrier. of alpha" emS loT Sde^dS] ?£* ZZT ^J^T^^ ^™ don in ic 28 . This expStion^Ses^u I M aSdVS '° n " SS2 U SrS^f?J^^?_ t » .Pe»t»t» a po^LKallarSer . first the problem of the one dimensional rectangular harrier will he considered. Classically an Incident particle of energy E (see Pig. II 1. 1), coming from the left/" could not penetrate into re -Ion II, since U > E. It would be completely reflected. However, Ufr) FIG- . Ill . 1 quantum mechanically, the oarticle wh+iMn ■*« ■ ia nerriqiiv -r^fit^^L.i ^ P ■ ' whieh ls an incident wave, par ' lalI; reflected ana partially transmitted. It *f ^^^particle travelling to the Let f =dp Li tl urn %^Q ■£&-£>*) is b.h'e reflected wave i{-kz-u&) and ^ ^/€ i-g the transmitted wave. The probability of penetration or the w» rtt3 ,. transm itted intensity HPT transparency - iHHHiHt"intenHt'y * l^r Considering the time Independent Dart of/ re 1 t &-, These are solutions of the time Independent wave equate d ■%*■ * In region II the wave equation becomes ^jjj-i ft* t'H c // —° Lr:t B ~U-E Then g„KgV*3* ^ L g-^- j, phfsicallV " — tn& WaYe -^tion must be well behaved Thus $ and || are cort: x - b. nuous across the boundaries x = and 55 56 Rectangular barrier Ch. Ill Thus 4- +/3= K +l since % (0) =$'{.*} m^ , c^ & )^ute MbM u - l 6 Eliminating a between the first two of these equations gives Solving the last two of these equations for K and L gives K-fe-T-yje- L .x £•"'-(" f)e « SJL Thus * = £ Since ■T - 1 *=„£*?, A^jjF ~, transparency ^ ^ for &»' ^JP^A III.l Notice that as h -*■ 0, this approaches the classical limit of zero transparency. Also the trans- parency is greater for particles of smaller mass. This explains why electrons have little trouble in going from one atom to another in diatomic molecules and metals. See PIG III. 2. tT/4.3 sw^ PIG HI, 2 Diatomic molecule with one electron B. Barrier of Arbitrary Shape The order of magnitude of the transparency of an arbitrarily shaped barrier can be obtained by finding the average "height" and treating as a rectangular barrier. PIG III.? Ch. Ill III.l then gives Rectangular barrier transparency e '^Mfa^l ^ 57 III. 2 Problem: sidered bump In 100 ft. get past actually the c ar A very slowly moving car of 1 ton (kinetic energy con- almosL zero) encounters a sinusoidal T , M the road which Is 1 ft. high and long. Classically, the car can't However 112,2 shows that there is a finite probability (w) that can overcome the bump. TJ(x) = jjg x ig x 2.54 3in uU-aJ_ 3050 f> U(x) dx ^ 5.3 x 10 J <? -lf.3*!0 ■it '10 -L.Z *t&~ .3$ In the followine figure, III .2 is used to calculate the transparency for Protons penetrating the coulomb barrier of nuclei ot A = 1, a, 90, anci 238. These are slotted as a function of troton energy divided by barrier energy. TVMHsP^ftFrfcr CUf(v£ A 2 a*(«Rtia? £T a, 1 I OSMev %■ $ 4 ui c w 4a U d 238 f& HJL Trans oar encv to Protons of Nuclear Coulomb Barriers 58 Alpha Decay Ch. Ill C, Serai-classical Application to Alpha Decay III. 2 will now be applied to calculate the probability of an UW alpha particle penetrating the potential barrier due to a nucleus. First, assump- tions must be made concerning this barrier. As an alpha particle Is brought from r=oo, the main interaction force is the electro- static repulsion which gives the potential U(r) =3Zgr . Let R be the distance where the alpha particle first "contacts" the nucleus. Then It encounters the strong short range attractive forces until it Is within the nucleus. Once the alpha part- pig III. 4 icle Is in the nucleus it is considered as a free particle. The depth of this well is not very great for the alpha active nuclei, since the lowest energy state "must give a positive erergy eigen-' value. The height of the barrier is much greater, however. For uranium it is 29 Mev if equation 1.4, page 6, is used for R. probability / sec. of escape = rate of hitting barrier x transparency or T^r— 1 , f "R~ x fc- e fe * where /j, is the re- duced mass of alpha particle and nucleus 7=r ^ tO since V lr/ v 10 and p-v/o - ' 1 where & = ,EJt ^jk*^f«TJ5) -sT) j ~'°" «gW*VjHg-4FF >j 111.5 For a large br-arrier G has the following approximate form: = \Tv / where V is the final velocity of the °L particle Problem : Using formula I. 8, page 7, for the masses, A net ermine which of the following nuclei are unstable loo" to alpha decay and determine their lifetimes. 150 62 For the first two nuclei listed, the mass of 200 80 the daughter exceeds the mass of the parent, hence £50 97 they are stable. Of the next three, Z=BO has the 300 113 longest lifetime and will be calculated in detail here, 11(200, 80) = lit 198, 7 8) + Ke 4 + $ M( 200, 80) =200. 04474 ■ , „ ,, M( 196, 78 ) =196.03826 Q a 14(200,80) - p,l( 196,7 8) + He^J He*- 4.00390 $ =- 0.00258 am = 3.84 x 10 -6 ergs zZe 2 - 3,84 x 10-°ergs b = 9.52 x 10-l^cm R -V = 1.5 x 10~ 15 {196) 3 = 8.7 x 10-l 3 cm bv 1.4 Alpha decay III. 3 gives for G: = 127 x 1,198 = 152 £ -v fa**^-**-* t - 3 x 1035 vra . since 3>15 ^ iq7 sec = ^ ^ ?hus this nucleus would appear stable, 59 T>. Virtual Level Theory of Alpha Decay Now a complete quantum mechanical rresen^Hn. , m v. given, includlrs- the , n i„f=A^ £ ., presentation will be nucleus. This lladsto h vtf,° 7^ ^ pha P"^°l« inside the Problem ; At 2 = there is an infinite barrier. The onlv finite solution In this region is zero since -Sp? %--■:■ £ I Let i>*= 'Jr JHE PIG III. 5 for x £ 0. In regions I and III d^ + ^ Since ^ must be zero in the Jnfi'nWa «** 4-- , ffiust be continuous. ml h*S infi - lte potential region anfl / + t>* #- e v e ha ye where the arbitrary constant Is chosen as one. ft ^e'^^e' < P&C-4-) I 60 Virtual level theory The continuity conditions at the boundaries give since *&.&>=*.# AJ Let = 2q(b-a) *V->c 4 = C : + e C£ _ & #>*> -g'^ 4^) - fei Ch, III III. 4 j&tfa pfiz _ & yj than S arV^Hr lntenslt y of ^ *•■ usually much greater than thaj of ^ . however we are interested In the case where r^ is aa snail as possible, II 1. 4 gives C + = iiLemps* +. ^£-A^ra^v . 5 T ~k- * * 1 The values of E which satisfy the above equation give the small- est possible intensity outside. These values of E are known as the virtual levels. It is seen the out- side intensity decreases very sharply In ilie energy region of the virtual level. renera! 1 sSuaMo^S fo }} ^ W* the first curve shows the §!£? - flvJ n Wh ^ n C+ f °' The second curve is for the vir- fo^si T f?l Th6 fi2ial curve shows ^e behavior of ^ lor an energy sllgntly greater than the virtual level energy Ch. Ill Virtual level theory This same virtual level approach will now be used with the W.K.B. method In solving the ppdblsBi of the potential barrier- experienced by tlie r.l.Tiiia nsrT,i cle . FIG III. 7 61 FIG III. 6 FIG 7 gives the potential barrier experienced by the alpha aruicle. Since this Is a three dimensional problem, spherical oordmates vfill be used which give II 1. 5 for the radial part part G of the wave function. The complete wave equation is Let $m t g^) = fjpq Y tr jw) than ^[a^)] +>[*$(£- i^f) - l^jMw =0 for regions II and III III. 5 First the case where the angular -omentum of the alpha narti cle with respect to tne nucleus Is zero (s-state, £= 0} shall bs considered. 0^Y^4 e >=* where «.=A,^0 for region I for real on II for region III III. 6 Since these are all positive quantit ies , T , g, & , and $ are all u x = sirjr since £f#* ^/^ must be finite 62 Virtual level theory Ch . ni The W.ff.B. method gives for the general solution in II a^o!; K r B : Tir^Jll^tT connectlng "«W***« *«"«. fce*?^ J'^^^^^Fj Thus « ^ t^i^^aM G Is defin ed by III. 3 s*£ This gives for u^ At large r %%*?** * J *t-tfee- t +4(?(r} III. 7 SlatedTTh^ V*" Chapter > *■«« 67 > C_ and G + will be cal- culated In the region near a virtual level with the result ^- ^W where fC^^J^(iXR)-£} JS***! J" 3 X* ener S^ reparation from the nearby virtual level and n Is tne separation of two adjacent virtual level's. Ill, 7 becomes v^ III. 9 where a*« -* Thus except for a small energy range about the virtual level £ , S£ t £\£ t 3&iI s^ 1 " 8 " th *» ttal ^° ^ £ ole . tw* -**$*+- oo while /(£) <*£ for i i ™ S 2<!r ° - I ' robttbl lity or zero amplitude Lll r_s :h. Ill Virtual level theory 63 E n . ^ U) ln IIIj a11 corresponding to the same energy vlrLS'ene'rpv ^he°dffi^ 5*** f Mr W 8 W*™ «*»* the virtual energy. The distribution will turn out to he such that S^lr.tj will decrease exponentially with + ime HXa r>l+Z * f In summary the procedure will be- C1) So V?r?jS°?eSl? 1S ^ fuActi ° nS ln 6ne ^ «*i°* ab0 ^ fE) Expansion of ^(r,0) in terms of these eigen function (3) Determination of ra^e of decrease of ^ ? ^ unctlona - (1) Actually there is a continuous spectrum of eigen values and func- tions. The problem will first be treated as one of discrete levels by limiting region III to r< L. Later L will be made to approach infinity, which approaches -che continuous spectrum situation The region is lifted to r < L by placing an infinite barrLr'at - - L - L - lS taken large enough so that: t5?*3 k= /*£*& »/^^ IZXVTo'A,'* ; ff60il ' el >' " «>»• •«« (of n h.U-,av. ■4 K ~ %J* ~ z £"^SiS^^ 7 ?.sa2r d - nornaiized " hioh 8i *«° » Now /<H.A.~ i III. 9 gives for the amulitude inside- amplitude inside %.W The shape of the function u„ up to r - d fqe P w fil i„ D „ Inl G al 8 y the% Same /i°? ^ ^^ 1-elYnear ^%™u2 1 v T. ±n hxg ti the smooth line represents un which has an E n slightly below the lowest virtual level, E , and the dotted line is for an enero-v slightly above S . Let f(r) (of amplitude one) represent all these u n 's for the region r< d. Then u n (r) if to for r< d III. 30 64 Virtual level theory Ch. Ill If ^ is the wavelength of u n outside: rft-^p'sss-l. i. «. i& ~ A- 3* IF /U -£. -i.2^^4.^ M*^i m "■** >t !> III. 11 III. 12 (2) Expansion of ¥ (r,0) in terms of u n . Up to nov^ f(r) has teen defined only for r<d. For r > d it will he defined = 0. Then^frjO) = ffr) so that the alpha particle has zero probabil- ity to he outside the nucleus at t = 0, since f(r) = for r > d c = _ J¥ by ill. 10 >£^ This summation over n approaches _>-£^« M»g-*+**£*& using u n fr) for r < d from III. 10 as L-»°° With the help of II I. 12 the variable of integration can be trans- formed: -t Since the integrand has large values only over a small range of n and n Is very large, n may be taken outside the integral. ^JLfey &*&$££. f^" j. 'tcK4 using III. 11 to eliminate n Ch * TI1 Virtual level theory where ^-f /g" 5 55 = tfer Q -, PO _ i.APL Since - W3 — t-LK^- ^ f5 ' cJ r,t ' decreaSQ s to ^ of /^ a (r,o) at ^ff£ ^r- which Is —-5- g S after substituting the values of A and B from page 62 III. 13 T ~—fk- ~€ 1Vhere E in = E - U Is the kinetic energy in. s;i.de the nucleus. ■ * since V^ JS5 W since Ipproachf S Same rSSUlt " II1,3 Which U3ed a «ffll-dlw a ic«i Comparison with Experiment order? 1 of Ja^i?*^™*^* ° h9 2 kS Wlth e *P s ™^ as far as lifetime mlv^fS r c °£ cerned ' T he theoretical value of the 103 Tir?hf HZ ^ th \ ex P e Cental by such a factor as 10 . Thus the formula Is not very useful for computing life- times however, slncetis very sensitive to the radiuf R which appears In the exponent, a rough knowledge of \ will ^ive a Such .ore exact knowledge of R. In fact the j^* t ^ g™.^ ~* D x 1U i0 A^ Is determined by averaging 'the radii nf ninho E B S B hl s C ect- e C R a « CITl u ted USi ^ thiS lh6 ^ Sli if aS? i^ It* l sec 1 tlon 6S.. No correction for the radius of the alpha particle is made In this determination. Be the gives two | Scent ^neXlt" °! d " ^ thS " neW " Whlch flif S S SoS 4.0 Kl,iS.,? l^ arS m ° re in favor toda ^ Lethe's nev values are much larger because he considers that for most o. .he time the alpha particle does not exist separately L the Alpha decay, experimental ^ .■-'.'■■ J. i f ' i.i :" J O '-.' .—.■:■ :■- .- ■■-. J 1 1. ~1 Ch. Ill point cf ohe Gamow theory. This cpcibaMUH „? main weak the alnha r^i-vin -it i-t i prooaoility of pre-existance of d " tor lb „ ( -R" I to 10 lb J which gives a smaller harrier n^ larger radius for the same lifetime? " " Another consideration which has oeen neglected thus far is the possl cility of the alpha particle being emitted with angular momentum. The same mathematical treatment may be used if j£ 1JW lg added to t £ potential s ~ as can he seen by III. 5. This will increase the harrier and naif -life. See FIG 9. The effect Is not very great since an a as large as 5 changes r only by a factor of 10. PIG III. 9 The Geiger-JIuttall Law lonJ^J^i ael £ er ^fl Nut tall discovered the exoerirre-ntal re- lationship between members of the same radio-aotSe^Sr wnere R is the range in air III. 14 For the 3 radio-active families this const. Is 10-42.3 for , araniuj:]1 faEllly 10-44.?. for thorium Since in the energvregion for alphafulo" iS ife^™ v3 „ gives a relationship between ^ and the velocity Tin', ™Ti o?V checks with that given by the" Gamow Seoly (II? 3)^ve/£e limited energy range which is observed. E. Alpha Kay Spectra „„.„ f Many al f*" a eailtters 3h0 ' a fine structure in the enerev spectrum of the emitted ainhaq. ™ j -m- k*-i ^ . Z e - ier &y tSsit^of'a^lvS'r- ^ °™ the0r ^ Sives thf ratio of^n- stZl L " 8 £g Gnergy SrOUP t0 that <* *he maximum energy <TE is the energy difference and E Q ia :.i., e maximum energy. The in- general +p P nri nf 1 ^,™TZ , " ilLL ^O J-* ^ne maximum ene l^^ V ^jL^°fioTtSi e rJ^r rt3 th V iS Nation. The in ing dlffereni^ab^^ particle within the nucleus. existance of uhe alpha Gii. Ill 1«J >% SO 1 . site* L - E 5.709 5.728 5.873 6.16 6.S Relative intensity 1.1$ O.S.4. 1.8$ 704, 27% F IG III .10 Alpha Spectra 67 pare nt daughter PIG- III. 11 FIG- 11 la an energy level diagram Illustrating the phenom- enon ol" discrete energy groups, <*, Is the maximum energy alpha. The excited nuclear states of the daughter decay rapidly by gamma emission to the ground state,, E , where hv = E-t - E , etc. The energies of some of these gamma rays have he en observed and do check. ThC Long Range Alpha Emission Two nuclei, RaC and ThC ' , give an alpha spectrum which contains low intensity groups of alphas up to 2 Mev ahove the main group. The intensities of these long range groups are from 10-4 to 10 -6 of the total Inten- sity, This is because both RaC and ThC are products of a beta disintegration which leaves them in excited states. These excited states have a choice of decaying by a gamma ray to the ground parent state, or else by alpha to the daughter nucleus. FIG 12 show3 the situation for ThC. Since the gamma lifetimes are on the order of 10 -5 times smaller than the alpha lifetimes for the ex- cited parent states, the long range alpha intensities are verv low. Half livesirf 3 s 10-7 sec, r, a io-i4 ?=M 1Q-17 Decay of ThC ' FIG III. 12 F. Appendix to Gh.III The calculation of C + and C_ as used In III. 8 will be made here. See- page 62. The continuity conditions across the bound- ary at r = R give since ***.&& = *h(H) ■■ 68 Appendix Ch. Ill At the virtual level E = E , C + = or ^w-k-JS0 which is less than unity .for an unstable nucleus Thus Y% -z;/n7f where n is an integer 111.15 since 3E ft.* This may also be expressed In terras of J E by using III. 15, bu thus <fr C + - If'// ■* III. 16